Lyapunov-stability for the sliding-mode control of a turbocharger

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turbocharger increases the air-pressure in the intake manifold and thus enhances the capability of .... pi ∈ (pi,min,pi,max]=(pa, 3000hPa]. (9) px ∈ (px,min,px ...
Lyapunov-stability for the sliding-mode control of a turbocharger subject to state constraints Jan Schreibeis1 , Kai Wulff 1∗ , Johann Reger1 and Jaime A. Moreno2

Abstract—We consider the robust control design for the boostpressure control of an exhaust-gas turbocharger of an internal combustion engine. We design a sliding-mode controller using the super-twisting algorithm. In our main contribution we present a thorough stability analysis based on two different Lyapunov functions, one for the reaching phase and one for the slidingphase. Additionally, we consider state-constraints for each of the control phases and guarantee that the state stays within prior defined bounds. Index Terms—Sliding-Mode Control, Lyapunov Functions, State Constraints, Nonlinear Systems, Turbocharger Systems, Diesel Engine

I. I NTRODUCTION In order to increase efficiency and to reduce emissions in modern combustion engines the use of turbocharger has ever been increasing. Besides the automotive applications, turbochargers can be found in a wide field of applications including building and agricultural machinery, locomotive and ship engines and also modern power systems like block thermal power plants or gas turbine systems. This may account for the continuing interest in research on the control of turbochargers, e.g. [1], [2], [3], [4], [5], [6], [7]. Figure 1 shows the main components of the airpath of a diesel engine. The remaining energy of the exhaust-gas drives the turbine of the turbocharger. The compressor of the turbocharger increases the air-pressure in the intake manifold and thus enhances the capability of carrying oxygen to the combustion. The turbine is equipped with a variable turbinegeometry (VTG) which allows to adjust the power that drives the turbine and thus to impact on the boost-pressure in the intake manifold pi . For such systems, the modeling and parameter-identification usually is a tedious and challenging task [8], [9], [10]. The process dynamics exhibits pronounced nonlinearities with substantial uncertainties. Therefore, sliding-mode control appears to be a suitable approach for devising a simple, but robust controller. To the best knowledge of the authors the first attempt to use a sliding-mode approach to the boost-pressure The authors kindly acknowledge support by the German Academic Exchange Service (DAAD) with financial means of The German Federal Ministry for Education and Research (BMBF) through the project “LyapunovFunction based Adaptive Sliding-Mode Control” and through funding from the European Union Horizon 2020 research and innovation program under Marie Skłodowska-Curie grant agreement No. 734832. 1 Control Engineering Group, Technische Universität Ilmenau, P.O. Box 10 05 65, D-98684, Ilmenau, Germany 2 Eléctrica y Computación, Instituto de Ingeniería, Universidad Nacional Autónoma de México, 04510 México D.F., Mexico ∗ Corresponding author: [email protected]

ambient air

compressor

intercooler

intake manifold

C turbo charger

diesel engine exhaust gas recirculation

exhaust

T turbine with VTG

exhaust manifold

Fig. 1. Schematic of the airpath with turbocharger

control is published in [11]. There the authors consider a firstorder sliding-mode controller and achieve good performance in conjunction with a state-observer. Attractivity of the slidingmanifold is guaranteed by the reaching law, however, there is no formal proof of stability of the dynamics on the slidingmanifold. In [12], [13] a multi-variable approach is considered using both, the exhaust-gas recirculation and VTG, to control the massflow in the input-manifold together with the exhaustgas pressure as suggested in [14]. In this contribution we consider the SISO-case where only the VTG-actuator is available for the boost-pressure control. For the design of the sliding-mode controller we adopt the approach proposed in [11]. Our main contribution is the thorough stability analysis of the design in both, the reaching phase as well as the sliding phase. We present a stability proof for the dynamics on the sliding-manifold designing a novel Lyapunov function with piecewise linear level-sets. The construction of the Lyapunov function is inspired by the design of invariant sets [15]. As a further contribution, we consider state constraints for each phase and ensure that the state remains within prior defined bounds. We design a supertwisting sliding-mode controller [16] which is known to reduce chattering and thus actuator wear. The paper is organized as follows. The subsequent section presents the well-known third-order model of the airpath and describes the control task and the respective state constraints. In Section III we describe the system in regular form and design the super-twisting controller and its sliding-surface. The main results are contained in Section IV. We use a recently proposed Lyapunov function to guarantee asymptotic stability of the sliding-manifold. In order to guarantee asymptotic stability while also obeying the state constraints we consider a piecewise linear Lyapunov function. The Lyapunov function is constructed by matching the boundary of the feasible statespace. We derive conditions that guarantee negative definite-

ness of the derivative of the Lyapunov function along the solution in the sliding-phase. Section V provides simulations of the control and discusses the impact of the so-far disregarded actuator dynamics as well as well as numerical evaluations of the invariance. We close with the conclusions in Section VI. II. M ODELING We shall recall the well-established model for the airpath given in [14], [17] with parameters given in [4]. The thirdorder model of the airpath is derived by considering the massflow balances in the respective volumes, expressed by RTi (Wci + Wxi − Wie ) (1) Vi RTx p˙x = (Wie + Wf − Wxi − Wxt ) (2) Vx 1 P˙c = (−Pc + ηm Pt ) (3) τ where pi denotes the boost-pressure in the input manifold, px the pressure in the output manifold and Pc the power of the compressor. Wci is the flow rate from the compressor to the input manifold. Wxi is the flow rate from the output manifold to the input manifold and describes the exhaust gas recirculation. Wie is the flow rate from the input manifold into the engine and Wxt is the flow rate from the output manifold to the turbine. The flow rates and the turbine power Pt can be determined as in [18] and are given by p˙i =

ηc Pc   , cp Ta pi µ − 1 px s   Aegr (uegr ) px 2pi pi √ Wxi = 1− , px px RTx p i nm V d , Wie = ηv 120RTi     pa Wxt = (c1 uvtg + c2 ) c3 −1 +c4 px r s   px Tr 2pi pa 1− , pr Tx px px   µ  pa . Pt = Wxt cp Tx ηt 1 − px Wci =

(4)

(5) (6)

(7) (8)

The actuating variable is the VTG-position uvtg in (7) which is subject to some first-order dynamics. In this paper we shall consider the nominal case where no exhaust gas recirculation is considered. A complete list of the parameters of the model can be found in Table I. For the formal description of the state-space we choose  x > = p i p x Pc which shall be subject to practical constraints. In [17] it was shown that the set {(pi , px , Pc ) : pi > pa , px > pa , Pc > 0} is invariant such that any trajectory starting in this region stays therein for all future time. For practical applications, however, this set might violate some of the physical limits of the

TABLE I M ODELPARAMETERS FROM [4] Description Value nm Wf Wci Wxi Wie Wxt R pi px pr pa Pc Pt Ti Tx Tr Ta Vi Vx Vd ηm ηc ηt ηv cp µ τ c1 c2 c3 c4

Engine speed Mass-flow fuel Mass-flow compressor Mass-flow EGR Mass-flow intake manifold Fuel rate exhaust manifold Specific gas constant Intake manifold pressure Exhaust manifold pressure Reference pressure Ambient pressure Compressor power Turbine power Intake manifold temperature Exhaust manifold temperature Reference temperature Ambient temperature Intake manifold volume Exhaust manifold volume Displacement volume Turbo mechanical efficiency Compressor isentropic efficiency Turbine isentropic efficiency Volumetric efficiency Specific heat at constant pressure Constant Time-constant of turbocharger VTG Parameter VTG Parameter VTG Parameter VTG Parameter

33.3 7.2 – – – – 287 – – 1013 1013 – – 313 509 298 298 0.006 0.001 0.002 98 61 76 87 1014.4 0.286 0.11 -0.136 0.176 0.4 0.6

Unit  −1   s −1  kgh   −1 kgh−1  kgh−1  kgh−1  kgh   Jkg−1 K−1 [hPa] [hPa] [hPa] [hPa] [kW] [kW] [K] [K] [K] [K]  3  m3   m3  m [%] [%] [%] [%]   Jkg−1 K−1 [−] [s] [−] [−] [−] [−]

turbocharger. Therefore we shall specify stronger constraints by introducing bounds for each state defining the feasible region X ⊂ R3 : pi ∈ (pi,min , pi,max ] = (pa , 3000hPa]

(9)

px ∈ (px,min , px,max ] = (pa , 6000hPa]

(10)

Pc ∈ (Pc,min , Pc,max ] = (0, 15kW] .

(11)

The control task in this contribution is to design a controller for the boost pressure pi such that the state constraints given above are met for all time. While we shall consider set-point control only in this contribution, it seems also possible to extend the proposed procedures for allowing tracking control. III. C ONTROL DESIGN The design procedure of a sliding-mode controller can be split into two steps. The first step is to design a switching function σ(x) which allows to adjust the dynamics of the system in sliding-mode. In the second step a sliding-mode controller is designed such that σ(x) = 0 is reached in finite time (reaching-mode) and remains there (sliding-mode). For simplifying the control design we shall disregard the actuator dynamics of uvtg . As (7) may be readily solved for uvtg we may consider the exhaust-gas mass flow Wxt as input variable. With u = Wxt as the actuating variable, the airpath model is of input-affine form and thus may be written as x˙ = f (x) + g(x)u .

(12)

For the design of a sliding-mode controller it is convenient to describe the system in the so-called regular form proposed in [19] and [20]: x ˜˙ 1 =f1 (˜ x1 , x ˜2 ) ˙x ˜2 =f2 (˜ x1 , x ˜2 ) + g2 (˜ x1 , x ˜2 )u

(13)

variable S1 (˜ x1 ) acts as the input of the reduced system in the sliding-mode and a linear state-feedback controller of the following form may be designed to stabilize it, i.e. S1 (˜ x1 ) = Ψ(p∗x ) − k1 (pi − p∗i ) − k2 (z − z ∗ ) .

(22)

(14)

Thus, the switching function for the system in regular form is

with x ˜1 ∈ R and x ˜2 ∈ R some transformed states. State-transformations x ˜1 = φ(x), x ˜2 = φn (x) with φ : Rn → Rn−1 and φn : Rn → R that bring the examined system into regular form may be obtained by solving the set of partial differential equations

σ(pi , z, px ) = Ψ(px )−Ψ(p∗x )+k1 (pi −p∗i )+k2 (z−z ∗ ) . (23)

n−1

∂φ(x) g(x) = 0 . ∂x

(15)

T We consider the following transformation: x ˜1 := pi z and x ˜2 := px with Z δ px Ω(p)dp (16) z := Pc + ζ pa where ηm cp Tx ηt δ := , τ

RTx ζ := , Vx

 Ω(px ) := 1 −

pa px

µ .

With further abbreviations ηv Vd RTi ηc ηV Vd Tx α := , β := , γ := 120RTi Vi cp Ta 120Vx Ti Z δ px β , Ψ(px ) := Φ(pi ) :=  µ Ω(p)dp , pi ζ pa − 1 pa we obtain the system in regular form: (17) (18) (19)

Note that Ω(px ) ∈ [0, 1] and Ψ(px )are monotonically increasing functions. While we are considering the system (17)-(19) in absolute coordinates, we shall use occasionally the following notation for brevity: ∆pi := pi −

,

∆px := px −

p∗x

,

∗

∆z := z − z ,

where p∗i , p∗x , z ∗ denote the respective set-points. With the system in regular form we choose the switching function to be of the form σ(˜ x1 , x ˜2 ) = S2 (˜ x2 ) − S1 (˜ x1 ) ,

(20)

with S1 : R2 → R and S2 : R → R to be defined. If S2 is invertible then we may solve (20) for x ˜1 . Substituting into (13)-(14), we obtain the dynamics on the sliding-manifold, i.e. for σ(˜ x1 , x ˜2 ) ≡ 0 we have  x ˜˙ 1 = f1 x ˜1 , S2−1 (S1 (˜ x1 )) . Now, choosing S2 (˜ x2 ) = Ψ(px )

v(˜ ˙ x) = − L2 sgn(σ(˜ x)) .

(25)

The positive parameters L1 and L2 may be used to specify how fast the switching surface is reached. Substituting the derivative of the switching function into (24) and solving the equation for the input yields the control law for the sliding mode controller γnm pi k1 p˙i + k2 z˙ u(˜ x) = + Wf + ζ δ Ω(px ) p (26) −L1 |σ(˜ x)| sgn(σ(˜ x)) + v(˜ x) , − δ Ω(px ) v(˜ ˙ x) = −L2 sgn(σ(˜ x)) . (27) IV. S TABILITY ANALYSIS

p˙i = −αnm pi + Φ(pi )(z − Ψ(px )) 1 δ z˙ = Ω(px )(γnm pi + ζWf ) − (z − Ψ(px )) ζ τ p˙x = γnm pi + ζWf − ζu .

p∗i

The second step is to find the sliding-mode control law. This can be done via the so called reaching law approach. The first step in applying the reaching law approach is to define the dynamics of the switching function in a way such that the switching surface is reached in finite time and that the system stays on it thereafter. For a super-twisting sliding-mode controller the dynamics results in p x)| sgn(σ(˜ x)) + v(˜ x) (24) σ(˜ ˙ x) = − L1 |σ(˜

(21)

A. Reaching Phase Using Lyapunov function approaches for the super-twisting algorithm, global stability may also be shown for some classes of bounded uncertainties. In [21] novel Lyapunov functions are proposed that guarantee reaching of the sliding-manifold in finite time. Defining p > > ξ = ξ1 ξ2 := |σ| sgn(σ) v system (24)-(25) may be rewritten as 1 ξ˙ = Aξ |ξ1 |

with

A=

 1 − 2 L1 −L2

1 2

0

 .

Consider the Lyapunov function candidate V (x) = ξ > P ξ . Since A is Hurwitz it may be shown that for every Q > 0 the Algebraic Lyapunov Equation A> P + P A = −Q has a unique solution P . In view of [21], this renders the right hand side of 1 > V˙ = − ξ Qξ |ξ1 | negative (almost everywhere) and ensures global asymptotic stability of σ = 0.

B. Sliding Phase It remains to be shown that the zero-dynamics σ(˜ x) ≡ 0 of system (17)-(19) with output (23) and control (26)-(27) is asymptotically stable while obeying state constraints (9)-(11). To this end, we calculate the boundary of the feasible region on the sliding-manifold Xσ = X ∩ {x|σ(x) = 0}. Substituting the extremal values of (9)-(11) into (23) and requiring σ = 0 reveals that the extremal values of Pc and Ψ depend on ∆pi :  k1 1 + k2 Ψ(p∗x ) − Ψ(px,min ) − ∆pi Pˆc (∆pi ) = Pc∗ + k2 k2  k1 1 + k 2 ∗ ∗ Pˇc (∆pi ) = Pc + Ψ(px ) − Ψ(px,max ) − ∆pi k2 k2 k k 2 1 ∗ ˆ Ψ(∆p ∆pi − (Pc,max − Pc∗ ) i ) = Ψ(px ) − 1 + k2 1 + k2 k2 k1 ∗ ˇ ∆pi − (Pc,min − Pc∗ ) . Ψ(∆p i ) = Ψ(px ) − 1 + k2 1 + k2

Fig. 2. Feasible region Xσ and estimated domain of attraction via quadratic Lyapunov function for p∗i = 1500hPa, k1 = 0, k2 = 0.1

The boundaries for state z on the sliding-manifold then read ( Pˆc (∆pi ) + Ψ(px,min ) if Pˆc (∆pi ) ≤ Pc,max region. Thus, we adopt the construction of a piecewise linear zmax (∆pi ) = Lyapunov function candidate, e.g. [22], [23], of the form ˇ Pc,max + Ψ(∆p if Pˆc (∆pi ) > Pc,max i) ( V (˜ x1 ) = max m> ˜1 i x i Pˇc (∆pi ) + Ψ(px,max ) if Pˇc (∆pi ) ≥ Pc,min zmin (∆pi ) = n ˆ Pc,min + Ψ(∆p if Pˇc (∆pi ) < Pc,min . where mi ∈ R are the outer normal vectors of the edges i) of the polygon describing the level sets. As V (˜ x1 ) is not and the feasible interval for pi remains the same. differentiable at the vertices of the polygon we consider the We choose the parameters k1 , k2 of the switching func- Dini-derivative instead: tion (23) such that the feasible region on the sliding-manifold V (˜ x1 + τ f1 (˜ x1 )) − V (˜ x1 ) D+ V (˜ x1 ) = lim sup . Xσ is invariant by considering each bound. Note that the τ + τ →0 boundary of Xσ is a polygon with four edges. The slope −k1 Along the edges of the polygon the Dini-derivative corre1 of the upper edge zmax (∆pi ) is either −k k2 or 1+k2 . Since the flow for pi = pa is perpendicular to the boundary we sponds to the classical derivative. At the vertices the Dinihave to choose k1 = 0. In light of that all bounds pi,min , derivative follows to be [24] pi,max , zmin , zmax are constant which yields a rectanguD+ V (˜ x1 ) = max m> x1 ) , (28) i f1 (˜ i∈J(˜ x1 ) lar shape for the feasible region Xσ . For the upper bound pi = pi,max we require p˙i ≤ 0. Substituting this into (17) where J(˜ x1 ) denotes the set of indices of active constraints  yields Pc,max ≤ α nm pi,max Φ(pi,max )−1 = 17471. Indeed, J(˜ x1 ) = i | m> x ˜1 = V (˜ x1 ) . i this holds such that there is no further requirement imposed For constructing a suitable Lyapunov function, matching on k2 . For the upper bound z = zmax , it can be shown that z˙ < 0 for k2 > 0. The lower bound z = zmin , however, yields with the feasible region Xσ , we choose the level sets as conditions depending on pi , p∗i , px,max , Pc,min . Numerical paraxial rectangles and the normal vectors on each face are > > > evaluations reveal that choosing k2 ∈ [0, 0.125] is suitable m> 1 = (1 0), m2 = (0 1), m3 = (−1 0), m4 = (0 − 1). Since the faces of the level sets are paraxial, the Dinifor the whole feasible region X . derivative is simply given by the (signed) flow of the two Figure 2 shows the feasible region Xσ (blue rectangle) on states. E.g. the Dini-derivative at the upper right vertex reads the sliding-manifold for the values k1 = 0, k2 = 0.1. For a comparison we calculated an estimate of the domain of D+ V (˜ x1 ) = max{p˙i , z} ˙ . attraction by considering a quadratic Lyapunov function based on a linearization of the nonlinear dynamics at the origin. The We now design two function that describe the evolution of the largest invariant set within the feasible region obtained by this vertices: fro describing the location of the upper right vertex Lyapunov function is depicted by the narrow (green) ellipse. It (∆pi,ro , ∆zro ) and flu describing the location of the lower is clear that a quadratic approach is by by no means ideal for left vertex (∆pi,lu , ∆zlu ). We require that obtaining the largest possible invariant set within the feasible fro : [0, ∆zmax ] → [0, ∆pi,max ] region for two reasons: The feasible region is a polygon and flu : [∆zmin , 0] → [∆pi,min , 0] is not symmetric with respect to the origin. We shall construct a Lyapunov function with piecewise strictly monotonically increasing for ∆z 6= 0, and fro (0) = linear level-sets to account for the properties of the feasible flu (0) = 0, fro (∆zmax ) = ∆pi,max , flu (∆zmin ) = ∆pi,min .

Fig. 3. Feasible region Xσ , limiting curves flp , flz (red), loci of vertices −1 −1 fro , flu , flu , fro (green) for p∗i = 1500hPa, k1 = 0, k2 = 0.1

In order to find conditions for fro , flu that render D+ V (˜ x1 ) negative, we consider the loci of points flp , flz in the state space with either p˙i = 0 or z˙ = 0. These limiting curves partition the state space into four regions (see Figure 3): Xσ+− with p˙i > 0, z˙ < 0 ,

Xσ−− with p˙i < 0, z˙ < 0 ,

Xσ++ with p˙i > 0, z˙ > 0 ,

Xσ−+ with p˙i < 0, z˙ > 0 .

Thus the upper right vertex of the level-set has to be within Xσ−− , the upper left vertex has to be within Xσ+− , the lower left vertex has to be within Xσ++ , the lower right vertex has to be within Xσ−+ . This is guaranteed whenever fro (∆z) ∈ Xσ−− flu (∆z) ∈

Xσ++

∀ ∆z ∈ [0, ∆zmax ]

(29)

∀ ∆z ∈ [∆zmin , 0] .

(30)

Fig. 4. Simulation of overall system with input dynamics (k1 = 0, k2 = 0.1)

∆p

∆z . For lower left vertices we choose flu (∆z) = ∆zi,min min the upper right vertices a linear combination does not proof suitable, thus we choose fro (∆z) = q2 ∆z 2 + q1 ∆z with q2 = 0.0347, q1 = 50. It is readily verified that the conditions (29), (30) hold and D+ V (˜ x1 ) < 0 for all x ˜1 ∈ Xσ \ {0}. Thus, the reduced-order system is asymptotically stable with region of attraction containing the complete feasible region.

The Lyapunov function then takes the following form:  −1  −1 −flu (∆pi ) fro(∆z) fro (∆pi ) −flu(∆z) V. S IMULATION RESULTS V (˜ x1 ) = max , , , |∆zmin | |∆pi,max | |∆zmax | |∆pi,min | For the validation of our design we conduct some simula(31) tions using the full nonlinear model (2)-(3) including a firstwhere the limiting curves are obtained from (17) and (18) with order lag for the actuator dynamics σ(˜ x) = 0 in (23), i.e. Tlag u˙ vtg,act + uvtg,act = uvtg αnm (p∗ ∗ i +∆pi ) − P c Ψ(p∗ i +∆pi ) flp (∆pi ) = (32) with Tlag = 0.104s, where uvtg,act ∈ [0, 1] denotes the 1 + k2 actual position of the turbocharger vanes. The actuator input! 1 ∗ signal uvtg is obtained using the control law (26)-(27) for Wxt (P + (1 + k )∆z) 1 2 c τ flz (∆z) = − ζWf − p∗i . and solving (7) for uvtg . δ −1 ∗ γnm ζ Ω(Ψ (Ψ(px ) − k2 ∆z)) Figure 4 shows the simulation of the closed-loop system for (33) a step from pi = 1500hPa to pi = 2000hPa. The transition Figure 3 shows the phase plane of the reduced system with takes roughly 0.5 seconds and all states remain within their the boundaries of the feasible region (blue), the limiting curves bounds. The fourth subplot shows the output of the superflp , flz (red) and the loci of the vertices of level sets (green) twisting controller and the applied vane position uvtg,act . for the equilibrium point p∗i = 1500hPa. For the loci of the Although the output of the super-twisting controller does not

R EFERENCES

Fig. 5. Phase portrait of the complete system.

chatter the rate of change is too fast for the vane position to track it. This results in a decreasing oscillation of the controller output. We conclude that the specific super-twisting controller stabilizes the desired set-point even under the presence of the unmodelled actuator dynamics while keeping the states within their boundaries. Simulations with a large variation of the time-constant Tlag show robustness of the stability against this unmodelled dynamics. Figure 5 shows a three dimensional phase portrait of the system with its state boundaries (black lines). Simulations show that this cuboid is not positive invariant since trajectories starting at high Pc violate the upper boundary PC,max and trajectories starting at low Pc violate the lower boundary PC,min . This effect is due to the considered actuator dynamics. Notwithstanding, it may be shown that by slightly restricting the admissible initial values of Pc we are able to generate a region for which all trajectories starting in this region do not leave the feasible region. This cuboid is also shown in Figure 5 with red, slightly transparent color. The blue trajectories start on the surface of the defined cuboid and show the behavior of the controlled system. All trajectories stay within the defined boundaries and, as shown in Section IV, converge to the sliding manifold, shown in yellow color. VI. C ONCLUSION We have devised a controller for the boost-pressure control in an exhaust-gas turbocharger system. Requiring the variable turbine geometry as the only control input, we have transformed the system into regular form. For the set-point control we chose the super-twisting algorithm and show Lyapunov stability for both the reaching phase and the sliding phase. In each phase the system state stays within the predefined admissible state bounds which, consequently, allows the safe operation. For this purpose, we present a novel stability proof using a piecewise linear Lyapunov function and propose the respective design procedure.

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